Question

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suppose that $\triangle pqr$ is isosceles with base $\overline{pq}$.
suppose also that $m\angle p = (4x + 2)^\circ$ and $m\angle r = (3x + 11)^\circ$.
find the degree measure of each angle in the triangle.
(there is an image of triangle pqr with angle p labeled $(4x + 2)^\circ$, angle r labeled $(3x + 11)^\circ$, and sides pr and qr marked as equal.)
$m\angle p = \square^\circ$
$m\angle q = \square^\circ$
$m\angle r = \square^\circ$

Explanation:

Step1: Identify equal angles

In isosceles triangle \( \triangle PQR \) with base \( \overline{PQ} \), the legs are \( PR \) and \( QR \), so \( \angle P = \angle Q \). Given \( m\angle P=(4x + 2)^\circ \), so \( m\angle Q=(4x + 2)^\circ \), and \( m\angle R=(3x + 11)^\circ \).

Step2: Use triangle angle sum

The sum of angles in a triangle is \( 180^\circ \). So, \( m\angle P + m\angle Q + m\angle R=180^\circ \). Substitute the angle measures: \( (4x + 2)+(4x + 2)+(3x + 11)=180 \).

Step3: Solve for \( x \)

Combine like terms: \( 4x+4x + 3x+2 + 2+11 = 180 \) \( \Rightarrow 11x+15 = 180 \). Subtract 15: \( 11x=165 \). Divide by 11: \( x = 15 \).

Step4: Find each angle

• \( m\angle P=(4x + 2)^\circ=(4\times15 + 2)^\circ = 62^\circ \)
• \( m\angle Q=m\angle P = 62^\circ \)
• \( m\angle R=(3x + 11)^\circ=(3\times15 + 11)^\circ = 56^\circ \)

Answer:

\( m\angle P = 62^\circ \), \( m\angle Q = 62^\circ \), \( m\angle R = 56^\circ \)